Nonlinear approximation of functions based on non-negative least squares solver

Abstract

In computational practice, most attention is paid to rational approximations of functions and approximations by the sum of exponents. We consider a wide enough class of nonlinear approximations characterized by a set of two required parameters. The approximating function is linear in the first parameter; these parameters are assumed to be positive. The individual terms of the approximating function represent a fixed function that depends nonlinearly on the second parameter. A numerical approximation minimizes the residual functional by approximating function values at individual points. The second parameter's value is set on a more extensive set of points of the interval of permissible values. The proposed approach's key feature consists in determining the first parameter on each separate iteration of the classical non-negative least squares method. The computational algorithm is used to rational approximate the function $x^{-\alpha}, \ 0 < \alpha < 1, \ x \geq 1$. The second example concerns the approximation of the stretching exponential function $\exp(- x^{\alpha} ), \ \ \quad 0 < \alpha < 1$ at $ x \geq 0$ by the sum of exponents.